The system (REF ) can be refined by replacing the variables \(\lambda _{x,a}\)  with variables \(\lambda _{S,f}\) , where \(S\)  is a set of vertices of \(\mathbf {X}\)  of size at most \(k\)  and \(f\)  is a function from \(S\)  to \(V(\mathbf {A})\) . Solving such refined system over the set of nonnegative rational numbers (resp. integer numbers) would then mean finding rational nonnegative (resp. integer) distributions over the set of partial assignments from portions of the instance of size at most \(k\)  to \(\mathbf {A}\) . The former choice results in the Sherali-Adams LP hierarchy 
[1]},
which we call the \(\operatorname{BLP}\)  hierarchy; the latter results in the affine integer programming hierarchy [2]}, which we call the \(\operatorname{AIP}\)  hierarchy.
Crucially, the former but not the latter choice ensures local consistency: Each assignment receiving nonzero weight in the \(\operatorname{BLP}\)  hierarchy corresponds to a partial homomorphism, while the same is not true for the \(\operatorname{AIP}\)  hierarchy. Equivalently, the \(\operatorname{BLP}\)  hierarchy is at least as strong as the bounded-width algorithm [3]}, [4]}, [5]} (and, in fact, strictly stronger, see [6]}). It is also worth noting that these hierarchies are still complete but not necessarily sound, and they become progressively stronger as the level \(k\)  increases. In particular, the \(\operatorname{BLP}\)  hierarchy is “sound in the limit”, in the sense that its \(k\) -th level correctly classifies all instances of size \(k\)  or less – which is clear from the fact that a partial homomorphism over the whole domain is a homomorphism.
